In this piece, I delve into some thoughts I\u27ve had about decision theory. These have been inspired by the vaccine rollout phase of the COVID-19 Pandemic. I focus on decision making under uncertainty, as it relates to the decision to get vaccinated or not

Lewis, Michael A.

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Journal of Humanistic Mathematics Volume 12 | Issue 1 January 2022 COVID-19, Vaccines, and Decision Theory Michael A. Lewis Hunter College School of Social Work Follow this and additional works at: https://scholarship.claremont.edu/jhm  Part of the Arts and Humanities Commons, Behavioral Economics Commons, and the Mathematics Commons Recommended Citation Michael A. Lewis, "COVID-19, Vaccines, and Decision Theory," Journal of Humanistic Mathematics, Volume 12 Issue 1 (January 2022), pages 210-216. DOI: 10.5642/jhummath.202201.15. Available at: https://scholarship.claremont.edu/jhm/vol12/iss1/15 ©2022 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. COVID-19, Vaccines, and Decision TheoryMichael A. LewisThe Silberman School of Social Work at Hunter College, New York City, USAmichael.a.lewis@hunter.cuny.eduSynopsisIn this piece, I delve into some thoughts I’ve had about decision theory. Thesehave been inspired by the vaccine rollout phase of the COVID-19 Pandemic. Ifocus on decision making under uncertainty, as it relates to the decision to getvaccinated or not.COVID-19, in addition to being a deadly pandemic, has also been a seminaron the significance of mathematics for understanding real-world events. Asexperts have tried to understand, and help the rest of us understand, what’sbeen going on, mathematical modeling, exponential growth, probability the-ory, Bayes theorem, and a host of other topics have gotten their due.As I write these lines, about 63% of the U.S. population is fully vaccinatedagainst COVID-19 (COVID) [6]. The U.S. is also in the midst of a recentsurge in COVID cases which has apparently been caused by the highly trans-missible Omicron variant of the virus. Since entering office, President Bidenhas pleaded with Americans who’re not yet vaccinated to get a shot (or two,as the case may be) [5]. Yet many people are very reluctant to get vaccinated.There’s evidence that some of this reluctance is due to partisanship, with ahigher proportion of Republicans than Democrats saying they don’t plan toget vaccinated [4]. But partisanship probably doesn’t explain all of it. That’sbecause there’s also evidence that vaccine hesitancy is due to concerns abouttheir safety. That is, some appear to believe that vaccines against COVIDmay prevent the worst of COVID by creating health problems of their own[4]. Apparently, these folks would rather take their chances with COVID,in a “devil you know” sort of way. As I’ve considered all this, I’ve thoughtabout another area of mathematics which is relevant to the COVID mess:decision theory.Journal of Humanistic Mathematics Volume 12 Number 1 (January 2022)Michael A. Lewis 211Decision theory is an area of study which focuses on how agents (naturalagents like people or artificial ones like computers) do or should make deci-sions. The centering of how agents do make decisions is called “descriptivedecision theory,” while the focus on how they should make them is referred toas “normative decision theory.” Another distinction decision theorists makeis between decisions made under certainty versus those under uncertainty [1].A decision under certainty is one where the agent knows for sure the outcomeof a given act. A person trying to decide whether to buy an ice cream conemight reasonably be modeled as facing a decision under certainty, assumingit makes sense to think that such a person knows for sure that handing moneyover to the cashier will definitely result in an ice cream cone being given tothem.A decision under uncertainty is where the agent doesn’t know for sure theoutcome of a given act. This is the situation those deciding whether to getvaccinated against COVID are facing.To get a bit more precise, in order to model a decision under uncertainty,the modeler needs to do at least four things: 1) specify who the agent isor agents are 2) specify the possible acts the agent/agents may choose toengage in, 3) specify the possible states of the world (that is, the uncertainoutcomes which may occur) and 4) specify the payoffs to the agent/agentsunder all possible combinations of acts and states. When it comes to manymodels of uncertainty, a fifth thing is done: probabilities are associated withthe possible states of the world. One of the most common ways of presentingmodels in decision theory is through the use of tables [1].In Table 1 below, I provide an example of all this by modeling the decisionfacing those trying to decide whether to get vaccinated, the agents in themodel. The possible acts are 1) get vaccinated or 2) don’t get vaccinated.The possible states of the world are 1) contract COVID or 2) don’t contractCOVID. The payoffs from the combinations of possible acts and states arein the cells of the table and, hopefully, self-explanatory. All of the statisticsthat we’ve kept on COVID, vaccine efficacy and effectiveness, etc. notwith-standing, I suspect that many people find it difficult to assign probabilitiesto the possible payoffs in the table. That is, I suspect that many are decidingwhether to get vaccinated, or not, without appealing to precise numbers rep-resenting their beliefs about the probabilities of contracting COVID, sufferingsome adverse health effect as a result of getting vaccinated, etc.212 COVID-19, Vaccines, and Decision TheoryContract COVID Don’t Contract COVIDGet Vaccinated Likely mild or asymp-tomatic COVID infec-tion, unlikely to be hos-pitalized or die from it,possible bad health ef-fects of vaccine (a)Possible bad health ef-fects resulting from vac-cine (b)Don’t Get Vaccinated Perhaps mild or asymp-tomatic COVID infec-tion but more likelyto be moderate or se-vere and more likely tobe hospitalized or diefrom it, no possible badhealth effects from vac-cine (c)No possible bad healtheffects resulting fromvaccine (d)Table 1: A Decision Model for Getting Vaccinated. The letters in parentheses will be usedbelow to refer to cell payoffs.When trying to model decisions under uncertainty where agents may find itdifficult to assign probabilities to various outcomes, decision theorists havecome up with a number of ways of representing how agents might, or should,make decisions in such situations.First, these theorists make assumptions about agents’ preferences. Given thetable above, for example, we might assume that agents prefer payoff (d) to(b), (b) to (a), and (a) to (c). That is, (d) > (b) > (a) > (c), where “>”stands for “is preferred to.” Even if you don’t agree with this preferenceranking, assume it for the sake of illustration.One of the concepts decision theorists use to represent how people might orshould make decisions is the “maximin criterion” [1]. This criterion says thata decision maker chooses, or should choose, the act that generates the largestpossible minimum payoff. Let’s see how maximin applies in this case.First, focus on the “get vaccinated” row. Since payoff (b) is preferred topayoff (a), (a) is the minimum possible payoff for someone who chooses to getvaccinated. Next, consider the “don’t get vaccinated” row. We see from thepreference ranking, (d) > (b) > (a) > (c), that payoff (d) is preferred to (c)Michael A. Lewis 213(I’m assuming transitivity of preferences, a common assumption in decisiontheory). So, (c) is the minimum possible payoff for someone who chooses notto get vaccinated.Now compare payoffs (a) and (c); since (a) is preferred to (c), (a) is the onewhich meets the maximin criterion. That is, payoff (a) is the largest possibleminimum one.Those of us who’ve chosen to get vaccinated may, at least implicitly, haveappealed to maximin in order to make our decisions. How might decisiontheory capture the decisions of those who’ve chosen not to get vaccinated?One possibility is the “maximax criterion” which says that agents should ordo chose the act which generates the largest possible maximum payoff [1].Looking back at our table, payoff (b) is preferred to (a) and (d) is preferredto (c). So, the maximum payoff of getting vaccinated is (b) and of not gettingvaccinated is (d). And since payoff (d) is preferred to (b), the largest possiblemaximum is payoff is (d). That is, the act which meets the maximax criterionis the choice not to get vaccinated.One thing I should point out is that, so far, I’ve assumed that decision makerscare only about payoffs to themselves. Obviously, the choice to get vaccinatedor not doesn’t just affect the person making the choice but impacts otherswho might come into contact with them as well; that’s what the conceptof “herd immunity” is all about. To the extent that my model is meant torepresent choices people are actually making and the extent to which peoplein the real world consider the effects of their vaccine choice on others, this isa serious limitation of the model.I said earlier that decision theory is partly normative and, therefore, addresseshow agents, when faced with uncertainty, should make decisions. Neithermaximin, maximax, nor any other decision-making criterion in situationssuch as this one has received unanimous consent among decision theorists.That is, as far as decision theory is concerned and assuming my model closelyapproximates the decision situation agents are currently facing, it isn’t clearthat those who’ve decided not to get vaccinated are behaving irrationally.Part of the decision situation agents are currently facing has to do withincentives for getting vaccinated. Such incentives affect the payoffs agentsreceive from different combinations of acts and states of the world.214 COVID-19, Vaccines, and Decision TheoryWhen COVID vaccines were first being rolled out, not many institutionsrequired people to get vaccinated. That has changed as some employers, col-leges, and universities are requiring vaccines or boosters. How might vaccinemandates change the decision situation people would face?Take a look at Table 2 below:Contract COVID Don’t Contract COVIDGet Vaccinated Likely mild or asymp-tomatic COVID infec-tion, unlikely to be hos-pitalized or die from it,possible bad health ef-fects of vaccine, avoidloss of job or opportu-nity to attend college(a)Possible bad health ef-fects resulting from vac-cine, avoid loss of jobor opportunity to at-tend college (b)Don’t Get Vaccinated Perhaps mild or asymp-tomatic COVID infec-tion but more likelyto be moderate or se-vere and more likely tobe hospitalized or diefrom it, no possible badhealth effects from vac-cine, loss of job or op-portunity to attend col-lege (c)No possible bad healtheffects resulting fromvaccine, loss of job oropportunity to attendcollege (d)Table 2: A Decision Model for Getting Vaccinated When There Is A Vaccine Requirement.The letters in parentheses will be used below to refer to cell payoffs.This is similar to Table 1 from earlier but differs by including the possibility,should one refuse to get vaccinated, of losing a job or the opportunity toattend college. That is, I’ve assumed that all public and private sector em-ployers, as well as all colleges, have required people to get vaccinated; thosewho didn’t get immunized wouldn’t be allowed to work in the formal economyor attend college. How might such requirements affect people’s vaccinationdecision?Michael A. Lewis 215Presumably, the hope of those advocating vaccine mandates is that the fol-lowing preference ordering is the one decision makers would be acting on:(b) > (a) > (d) > (c). This would mean that if one chose vaccination,(b) > (a), and if they chose non-vaccination, (d) > (c). Since (a) > (c),those deciding on the basis of the maximin criterion would choose to get vac-cinated. Given that (b) > (d), people deciding on the basis of the maximaxcriterion would also choose vaccination. That’s because such folks, in a worldwhere they didn’t end up with COVID, would rather face the possible nega-tive health consequences of a vaccine but keep their jobs or the opportunityto go to college, than avoid such health consequences but lose the chance ata college education or livelihood.Whether a wave of vaccination mandates will ripple across the country re-mains to be seen. If so, as someone interested in decision theory, I’ll bewatching closely.References[1] Erik Angner, A Course in Behavioral Economics, Palgrave, UnitedKingdom, 2016.[2] Michael A. Lewis, Social Workers Count: Numbers and Social Issues,Oxford University Press, 2018.[3] Michael A. Lewis, “Are Logic and Math Relevant to Social Debates,”Journal of Humanistic Mathematics, Volume 10 Issue 1 (January 2020),pages 383-401. doi:10.5642/jhummath.202001.21[4] Cara Murez, “Poll Reveals Who’s Most Vaccine Hesitant and Why,”Web MD, April 29, 2021; available at https://www.webmd.com/vaccines/covid-19-vaccine/news/20210429/poll-reveals-whos-most-vaccine-hesitant-in-america-and-why, retrieved 8/3/2021.[5] Reuters, “Biden Urges Americans Please, Please, get Vaccinated,” on-line news story, July 19, 2021; available at https://www.reuters.com/world/us/biden-urges-americans-please-please-get-vaccinated-2021-07-19/, retrieved 8/3/21.[6] USA Facts, “US Coronavirus Vaccine Tracker,” online fact tracker, 2021;available at https://usafacts.org/visualizations/covid-vaccine-tracker-states/, retrieved 12/30/21.216 COVID-19, Vaccines, and Decision TheoryAuthor bio:I’m a social worker and sociologist by training; I have a Masters’ degree in social workand a PhD in sociology. By accident, I became interested in applied mathematics. Iwanted to learn economics, realized that I didn’t have the math background to do so,and taught myself the math I needed to know. It would be a huge stretch for me to callmyself a mathematician, applied or otherwise. I have a friend who is a well-known appliedmathematician, and I simply cannot do what he can. But that hasn’t stopped me fromloving mathematics and trying to apply it as much as my skill level and the social workschools where I’ve taught have allowed me to get away with [2, 3].

COVID-19, Vaccines, and Decision Theory

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COVID-19, Vaccines, and Decision Theory

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